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Definition df-card 7588
Description: Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 8184 for its value, cardval2 7640 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8189. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
df-card  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-card
StepHypRef Expression
1 ccrd 7584 . 2  class  card
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
4 vy . . . . . . 7  set  y
54cv 1631 . . . . . 6  class  y
62cv 1631 . . . . . 6  class  x
7 cen 6876 . . . . . 6  class  ~~
85, 6, 7wbr 4039 . . . . 5  wff  y  ~~  x
9 con0 4408 . . . . 5  class  On
108, 4, 9crab 2560 . . . 4  class  { y  e.  On  |  y 
~~  x }
1110cint 3878 . . 3  class  |^| { y  e.  On  |  y 
~~  x }
122, 3, 11cmpt 4093 . 2  class  ( x  e.  _V  |->  |^| { y  e.  On  |  y 
~~  x } )
131, 12wceq 1632 1  wff  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
Colors of variables: wff set class
This definition is referenced by:  cardf2  7592  cardval3  7601
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